Real Numbers and Elements of Number Theory Flashcards Preview

SAT Math Fundamentals > Real Numbers and Elements of Number Theory > Flashcards

Flashcards in Real Numbers and Elements of Number Theory Deck (61)
Loading flashcards...
1

What is the base of our numerical system?

Our numerical system is a decimal or base ten system. It uses digits from 0 to 9 as a base.

Our numerical system is a place-value system. This means that the place or location of a numeral determines its numerical value.

2
Name:

subsets of real numbers

The following are subsets of real numbers:

  • Natural numbers
  • Whole numbers
  • Integers
  • Rational numbers
  • Irrational numbers

3
Define:

natural numbers

Natural numbers are the set of counting numbers.

{1, 2, 3, 4, 5...}

Natural numbers are comprised of odd and even numbers.

The smallest natural number is 1; the largest natural number is infinity.

4
Define:

whole numbers

Whole numbers are the set of natural (counting) numbers and zero.

{0, 1, 2, 3, 4, 5...}

Whole numbers are comprised of odd and even numbers.

5
Define:

integers

Integers are the set of natural numbers, their negative opposites, and zero.

{...-3, -2, -1, 0, 1, 2, 3...}

Integers are comprised of whole numbers and the opposites of natural numbers.

6
Define:

rational numbers

Rational numbers are the numbers that can be expressed as simple fractions of two integers -- i.e. as ratios.

*** The denominator in the fraction cannot be zero.

Examples: 

5 = 5/1                   1.75 = 7/4

Rational numbers consist of integers and non-integral numbers (numbers that have terminating or repeating decimals).

7
Define:

irrational numbers

Irrational numbers are the numbers that cannot be written as terminating or repeating decimals.

Example:

For the purposes of the SAT, the most important irrational numbers are the square root of 2, the square root of 3, and Pi.

8
Define:

even numbers

 

A number that is divisible by 2 is called an even number.

{...-4, -2, 0, 2, 4...}

All numbers ending in 0, 2, 4, 6, and 8 are even.

9
Define:

odd numbers

A number that is not divisible by 2 is called an odd number.

{...-5, -3, -1, 1, 3, 5...}

All numbers ending in 1, 3, 5, 7, and 9 are odd.

10

Is the sum of two even numbers even or odd?

EVEN + EVEN = ?

EVEN + EVEN = EVEN

Example:

10 + 2 = 12

11

Is the difference between two even numbers even or odd?

EVEN - EVEN = ?

EVEN - EVEN = EVEN

Example: 

10 - 2 = 8

12

Is the sum of two odd numbers odd or even?

ODD + ODD = ?

ODD + ODD = EVEN

Example:

5 + 5 = 10

13

Is the difference between two odd numbers odd or even?

ODD - ODD = ?

ODD - ODD = EVEN

Example: 

5 - 3 = 2

14

Is the sum of an odd number and an even number odd or even?

EVEN + ODD = ?

EVEN + ODD = ODD

ODD + EVEN = ODD

Examples: 

4 + 3 = 7

5 + 4 = 9

15

Is the difference between an odd number and an even number odd or even?

EVEN - ODD = ?

ODD - EVEN = ?

EVEN - ODD = ODD

ODD - EVEN = ODD

Examples:

6 - 5 = 1

7 - 2 = 5

16

Is the product of two even numbers odd or even?

EVEN x EVEN = ?

EVEN x EVEN = EVEN

Example:

6 x 8 = 48

17

Is the product of two odd numbers odd or even?

ODD x ODD = ?

ODD x ODD = ODD

Example:

3 x 7 = 21

18

Is the product of an odd number and an even number even or odd?

EVEN x ODD = ?

EVEN x ODD = EVEN

Example:

6 x 3 = 18

*** When dividing odd or even numbers, the result can be a fraction, which is not a whole number; therefore, it is neither even nor odd.

19

When you raise even numbers to odd powers, is the result odd or even?

(EVEN)ODD = ?

(EVEN)ODD = EVEN

Example:

25 = 32

20

When you raise even numbers to even powers, is the result odd or even?

(EVEN)EVEN = ?

(EVEN)EVEN = EVEN

Example:

44 = 256

21

When you raise odd numbers to odd powers, is the result odd or even?

(ODD)ODD = ?

(ODD)ODD = ODD

Example:

33 = 27

22

When you raise odd numbers to even powers, is the result odd or even?

(ODD)EVEN = ?

(ODD)EVEN = ODD

Example:

72 = 49

23

True or False?

Any operation (addition, subtraction, multiplication, division or raising to power) on even numbers with another even number will result in an even number answer.

True.

If you understand that any two even numbers are divisible by 2, then logically the sum, the difference, the product, the quotient, the power of the two will always be divisible by two.

24

True or False?

Any operation (addition, subtraction, multiplication, division or raising to power) on odd numbers will result in an odd number.

False.

  • The sum and the difference of two odd #'s are even
  • The product, the quotient, and the power is odd

Think of ODD numbers as EVEN + 1. Or remind yourself that odd numbers end in 1, 3, 5, 7, or 9.

Example:

ODD + ODD = EVEN + EVEN + 2 = EVEN.

25

How do you express an odd number in terms of an even number?

ODD = EVEN + 1

Example:

ODD + EVEN = EVEN + EVEN + 1

26

How should you use number facts like ODD + ODD = EVEN on the SAT test?

You don't have to memorize them, but you have to be able to see that some questions may need you to recall and connect these facts to solve quickly.

Remember, SAT type questions often use simple facts, and the trick is seeing through them for a quick solution.

27

What are consecutive numbers?

Consecutive numbers follow the natural order and differ by 1.

{...4, 5, 6, 7, 8, 9...}

Consecutive even and consecutive odd numbers differ by 2.

{...2, 4, 6, 8...}

{...3, 5, 7, 9...}

28

In a set of consecutive integers, how do you find the number of integers between the smallest and the largest numbers, inclusively?

To count consecutive integers, subtract the smallest from the largest and add 1.

Example:

Count the integers from 14 to 51.

54 - 14 + 1 = 41

29

What type of number do you get as a result of adding different consecutive positive odd numbers?

1 + 3 = ?

1 + 3 + 5 + 7 + 9 = ?

The sum is a perfect square of the number of numbers being added together.

1 + 3 = 4 = 22

1 + 3 + 5 + 7 + 9 = 25 = 52

30
Define:

prime numbers

A prime number is a natural number greater than 1 whose only factors are itself and 1

The following is a set of prime numbers less than 100:

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}